Methods and apparatus to determine a twist parameter and/or a bend angle associated with a multi-core fiber

ABSTRACT

A multi-core fiber includes multiple optical cores, and for each different core of a set of different cores of the multiple optical cores, a total change in optical length is detected. The total change in optical length represents an accumulation of all changes in optical length for multiple segments of that different core up to a point on the multi-core fiber. A difference is determined between the total changes in optical length for cores of the set of different cores. A twist parameter and/or a bend angle associated with the multi-core fiber at the point on the multi-core fiber is/are determined based on the difference.

PRIORITY APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.17/142,634, filed Jan. 6, 2021, which is a continuation of U.S. patentapplication Ser. No. 16/908,414, filed Jun. 22, 2020, which is acontinuation of U.S. patent application Ser. No. 16/723,824, filed Dec.20, 2019, which is a continuation of U.S. patent application Ser. No.16/506,998, filed Jul. 9, 2019, which is a continuation of U.S. patentapplication Ser. No. 15/698,707, filed Sep. 8, 2017, which is acontinuation of U.S. patent application Ser. No. 14/326,004, filed Jul.8, 2014, which is a continuation of U.S. patent application Ser. No.12/874,901, filed Sep. 2, 2010, which claims priority from U.S.provisional patent applications 61/350,343, filed on Jun. 1, 2010,61/255,575, filed on Oct. 28, 2009, and 61/243,746, filed on Sep. 18,2009, the contents of which are incorporated herein by reference.

TECHNICAL FIELD

The technical field relates to optical measurements.

BACKGROUND

Shape measurement is a general term that includes sensing a structure'sposition in three dimensional space. This measurement coincides withwhat the human eye perceives as the position of an object. Since theeyes continually perform this task, one might assume that themeasurement is simple. If one considers a length of rope, one canphysically measure the position at every inch along the rope to estimatethe shape. But this task is tedious and is increasingly difficult withmore complex shapes. Another consideration is how to perform themeasurement if the rope cannot be physically reached or seen. If therope is contained within a sealed box, its position cannot be determinedby conventional measurement techniques. The rope in this example can bereplaced with an optical fiber.

Sensing the shape of a long and slender deformed cylinder, such as anoptical fiber, is useful in many applications ranging for example, frommanufacturing and construction to medicine and aerospace. In most ofthese applications, the shape sensing system must be able to accuratelydetermine the position of the fiber, e.g., within less than one percentof its length, and in many cases, less than one tenth of one percent ofits length. There are a number of approaches to the shape measurementproblem, but none adequately addresses the requirements of mostapplications because they are too slow, do not approach the requiredaccuracies, do not function in the presence of tight bends, or fail toadequately account for twist of the fiber. In many applications, thepresence of torsional forces that twist the fiber undermine theaccuracy, and thus, usefulness of these approaches.

Conventional approaches to measuring the shape of a fiber use strain asthe fundamental measurement signal. Strain is a ratio of the change inlength of a fiber segment post-stress verses the original length of thatsegment (pre-stress). As an object like a fiber is bent, material on theoutside of the bend is elongated, while the material on the inside ofthe bend is compressed. Knowing these changes in local strain andknowing the original position of the object, an approximation of the newposition of the fiber can be made.

In order to effectively sense position with high accuracy, several keyfactors must be addressed. First, for a strain-based approach, thestrain measurements are preferably accurate to tens of nanostrain (10parts per billion) levels. But high accuracy strain measurements are notreadily attainable by conventional resistive or optical strain gauges.Therefore, a new technique to measure the strain to extremely highaccuracy must be devised that is not strain-based in the conventionalsense.

Second, the presence of twist in the optical fiber must be measured to ahigh degree of accuracy and accounted for in the shape computation. Bycreating a multi-core fiber that is helixed and has a central core, thetwist of a fiber can be sensed. But the problem is how to obtain anaccuracy of rotational position better than 1 degree. For a highaccuracy rotational sensor, the position of strain sensors along thelength of the fiber must also be known to a high degree of accuracy.Therefore, some way of measuring the rotation rate of the outer cores inthe helixed fiber is desirable, which can then be used to correct thecalculation of the fiber position.

Third, fiber with multiple cores that is helixed at a sufficient rateand with Bragg gratings (a conventional optical strain gauge) isdifficult and expensive to make. It is therefore desirable to provide amethod of achieving nanostrain resolutions without Bragg gratings.

Fourth, multi-core fiber is typically not polarization-maintaining, andso polarization effects are preferably considered.

SUMMARY

The technology described below explains how to use the intrinsicproperties of optical fiber to enable very accurate shape calculation inlight of the above factors and considerations. In essence, the fiberposition is determined by interpreting the back reflections of laserlight scattered off the glass molecules within the fiber. Thismeasurement can be performed quickly, with a high resolution, and to ahigh degree of accuracy.

A very accurate measurement method and apparatus are disclosed formeasuring position and/or direction using a multi-core fiber. A changein optical length is detected in ones of the cores in the multi-corefiber up to a point on the multi-core fiber. A location and/or apointing direction are/is determined at the point on the multi-corefiber based on the detected changes in optical length. The pointingdirection corresponds to a bend angle of the multi-core fiber at theposition along the multi-core fiber determined based on orthonormalstrain signals. The accuracy of the determination is better than 0.5% ofthe optical length of the multi-core fiber up to the point on themulti-core fiber. In a preferred example embodiment, the determiningincludes determining a shape of at least a portion of the multi-corefiber based on the detected changes in optical length.

The determination may include calculating a bend angle of the multi-corefiber at any position along the multi-core fiber based on the detectedchanges in length up to the position. Thereafter, the shape of themulti-core fiber may be determined based on the calculated bend angle.The bend angle may be calculated in two or three dimensions.

Detecting the change in optical length preferably includes detecting anincremental change in optical length in the ones of the cores in themulti-core fiber for each of multiple segment lengths up to a point onthe multi-core fiber. The overall detected change in optical length isthen based on a combination of the incremental changes. The change inoptical length is determined by calculating an optical phase change ateach segment length along the multi-core fiber and unwrapping theoptical phase change to determine the optical length.

More specifically, in a non-limiting example embodiment, a phaseresponse of a light signal reflected in at least two of the multiplecores from multiple segment lengths may be detected. Strain on the fiberat the segment lengths causes a shift in the phase of the reflectedlight signal from the segment lengths in the two cores. The phaseresponse is preferably continuously monitored along the optical lengthof the multi-core fiber for each segment length.

In another non-limiting example embodiment, a reflected Rayleigh scatterpattern in the reflected light signal is detected for each segmentlength, thereby eliminating the need for Bragg gratings or the like. Thereflected Rayleigh scatter pattern is compared with a reference Rayleighscatter pattern for each segment length. The phase response isdetermined for each segment length based on the comparison.

A non-limiting example embodiment also determines a twist parameterassociated with the multi-core fiber at a point on the multi-core fiberbased on the detected changes in optical length of the multi-core fiber.The location at the point on the multi-core fiber is then translated toan orthonormal coordinate system based on the determined twistparameter. Preferably, the determined twist parameter is corrected foreach of the segment lengths.

In one example application where the multi-core fiber includes threeperipheral cores spaced around a fourth core along the center of themulti-core fiber, a phase response of a light signal reflected in eachof the four cores from each segment length is determined. Strain on themulti-core fiber at one or more of the segment lengths causes a shift inthe phase of the reflected light signal in each core. The phaseresponses for the three peripheral cores are averaged. The averagedphase response is combined with the phase response of the fourth core toremove a common mode strain. The twist parameter is then determined fromthe combined phase response.

In another non-limiting example embodiment, bend-induced optical lengthchanges along the multi-core fiber are determined and accounted for whendetermining the twist parameter. A bend at one of the segment lengths iscalculated and squared. The squared bend is multiplied by a constant toproduce a bend product which is combined with the determined change inoptical length of an outer core of the multi-core fiber at the onesegment length. One example beneficial application for this embodimentis for bend radii less than 50mm.

Another non-limiting example embodiment determines a rotationalorientation of the multi-core fiber about its axis at a point on themulti-core fiber at each of the segment lengths. A correction is madefor the effect of torsion and the resulting twist on the determinedorientation based on the detected changes in optical length of themultiple fiber cores. This correction is required to compute the correctbend direction.

Given a multi-core fiber characterized by a nominal spin rate, anothernon-limiting example embodiment determines an angular rotation of themulti-core fiber at a point on the multi-core fiber at each of thesegment lengths compared to the nominal spin rate of the multi-corefiber. A variation in the nominal spin rate at the point along themulti-core fiber is determined and corrected for. A “wobble factor” isdetermined for the multi-core fiber by constraining the multi-core fiberto a curved orientation in one plane. Correction is then made for thewobble factor when determining the location at the point on themulti-core fiber based on the detected changes in optical length.

In another non-limiting example embodiment, light is transmitted with atleast two polarization states along the multi-core fiber. Reflections ofthe light with the at least two polarization states are combined andused in determining the location or the pointing direction at the pointon the multi-core fiber based on the detected changes in optical length.The two polarization states include a first polarization state and asecond polarization state which are at least nominally orthogonal. Apolarization controller is used to transmit a first light signal at thefirst polarization state along the multi-core fiber and to transmit asecond light signal at the second polarization state along themulti-core fiber. A polarization-independent change in optical length ineach one of multiple cores in the multi-core fiber is calculated up tothe point on the multi-core fiber using reflections of the first andsecond light signals.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a cross-section of an example multi-core fiber;

FIG. 2 shows a bent multi-core fiber;

FIG. 3 shows that the bend in the fiber is proportional to the strain inthe off-center cores;

FIG. 4 shows that bend angle at any location along the fiber can bedetermined by a summation of all previous angles;

FIG. 5 shows that as a fiber containing Bragg gratings is strained, aphase difference measured from a reference state begins to accumulate;

FIG. 6 shows a clock that helps to visualize the relationship betweenthe phase shift and position;

FIG. 7 illustrates how a lack of resolution in measuring phase can beproblematic;

FIG. 8 is a graph that shows a phase difference of a Rayleigh scattersignal between a reference scan and a measurement scan at the beginningof a section of fiber that is under tension;

FIG. 9 is a graph that shows that coherence is lost with the referencemeasurement at a greater distance down a fiber under tension;

FIG. 10 shows optical phase plotted against frequency for two differentdelays;

FIG. 11 shows a recovered phase over a section of fiber where a third ofan index shift has occurred;

FIG. 12 illustrates an example of helically-spun multi-core shapesensing fiber;

FIG. 13 illustrates a non-limiting, example test multi-core opticalfiber;

FIG. 14 illustrates a cross-section of a helixed fiber where theposition of the outer cores appears to rotate around the center coreprogressing down the length of the fiber;

FIG. 15 is a graph that illustrates an example of variations in the spinrate of a fiber;

FIG. 16 is a graph that shows an example wobble signal with a periodicphase variation from a manufactured spin rate along the length of ashape sensing fiber;

FIG. 17 shows how torsion changes the spin rate of a shape sensing fiberbased on orientation of the force to the nominal spin direction of thefiber;

FIG. 18 shows an outer core that experiences twist modeled as aflattened cylinder as it translates along the surface;

FIG. 19 is a flowchart illustrating non-limiting example procedures tocalculate external twist along the fiber;

FIG. 20 shows an example data set for a generic shape that illustratesthe FIG. 19 procedures in more detail;

FIG. 21 is a graph that shows a slight deviation between the two phasecurves;

FIG. 22 is a graph illustrating a twist signal produced from FIG. 21 ;

FIG. 23 illustrates the necessity of compensating for twist in the shapecalculation;

FIG. 24 depicts example orthogonal strain curves for a fiber placed inseveral bends that all occur in the same plane;

FIG. 25 shows a flowchart diagram describing non-limiting, example stepsfor calculating shape from strain;

FIG. 26 illustrates that if each of multiple pointing vectors is placedhead-to-tail an accurate measurement of the shape results;

FIG. 27 is a non-limiting, example optical position and shape sensingsystem;

FIG. 28 is flowchart diagram illustrating non-limiting, example stepsfor calculating birefringence correction;

FIG. 29 shows a bend-induced strain profile of a cross section of ashape sensing fiber;

FIG. 30 shows two phase plots comparing a center core phase signal to anaverage phase of the outer cores;

FIG. 31 shows an example strain response for an outer core for a 40 mmdiameter fiber loop;

FIG. 32 is a graph showing a bend-induced birefringence correction forthe 40mm diameter fiber loop;

FIG. 33 is a graph comparing a twist signal with and without a 2n^(d)order birefringence correction;

FIG. 34 shows a non-limiting, example loop polarization controllerbetween a shape sensing fiber and a position and shape sensing system;

FIG. 35 shows an in-plane signal for a relatively simple shape where 1.4meters of shape sensing fiber is routed through a single 180 degree turnwith a bend radius of 50 mm;

FIG. 36 shows three successive out-of-plane measurements where betweeneach measurement, the polarization is varied using a polarizationcontroller;

FIG. 37 is a graph showing an example that two successive measurementsof the center core, with different input polarization states do not havea significant variation in phase response;

FIG. 38 is a graph showing an example that two successive measurementsof an outer core respond differently to input polarization providingevidence for birefringence in the shape sensing fiber;

FIG. 39 is a graph showing that correcting for birefringence improvedthe precision of the system; and

FIG. 40 is a graph showing that correcting for both first and secondorder birefringence improved the accuracy and precision of the system.

DETAILED DESCRIPTION

In the following description, for purposes of explanation andnon-limitation, specific details are set forth, such as particularnodes, functional entities, techniques, protocols, standards, etc. inorder to provide an understanding of the described technology. It willbe apparent to one skilled in the art that other embodiments may bepracticed apart from the specific details disclosed below. In otherinstances, detailed descriptions of well-known methods, devices,techniques, etc. are omitted so as not to obscure the description withunnecessary detail. Individual function blocks are shown in the figures.Those skilled in the art will appreciate that the functions of thoseblocks may be implemented using individual hardware circuits, usingsoftware programs and data in conjunction with a suitably programmedmicroprocessor or general purpose computer, using applications specificintegrated circuitry (ASIC), and/or using one or more digital signalprocessors (DSPs). The software program instructions and data may bestored on computer-readable storage medium and when the instructions areexecuted by a computer or other suitable processor control, the computeror processor performs the functions.

Thus, for example, it will be appreciated by those skilled in the artthat block diagrams herein can represent conceptual views ofillustrative circuitry or other functional units embodying theprinciples of the technology. Similarly, it will be appreciated that anyflow charts, state transition diagrams, pseudocode, and the likerepresent various processes which may be substantially represented incomputer readable medium and so executed by a computer or processor,whether or not such computer or processor is explicitly shown.

The functions of the various elements including functional blocks,including but not limited to those labeled or described as “computer”,“processor” or “controller” may be provided through the use of hardwaresuch as circuit hardware and/or hardware capable of executing softwarein the form of coded instructions stored on computer readable medium.Thus, such functions and illustrated functional blocks are to beunderstood as being either hardware-implemented and/orcomputer-implemented, and thus machine-implemented.

In terms of hardware implementation, the functional blocks may includeor encompass, without limitation, digital signal processor (DSP)hardware, reduced instruction set processor, hardware (e.g., digital oranalog) circuitry including but not limited to application specificintegrated circuit(s) (ASIC), and (where appropriate) state machinescapable of performing such functions.

In terms of computer implementation, a computer is generally understoodto comprise one or more processors or one or more controllers, and theterms computer and processor and controller may be employedinterchangeably herein. When provided by a computer or processor orcontroller, the functions may be provided by a single dedicated computeror processor or controller, by a single shared computer or processor orcontroller, or by a plurality of individual computers or processors orcontrollers, some of which may be shared or distributed. Moreover, useof the term “processor” or “controller” shall also be construed to referto other hardware capable of performing such functions and/or executingsoftware, such as the example hardware recited above.

Phase Tracking for Increased Angular Accuracy

FIG. 1 shows a cross-section of an example multi-core fiber 1 thatincludes a center core 2 and three peripheral cores 3, 4, and 5surrounded by coating 6. These cores 3-5 shown in this example arespaced apart by approximately 120 degrees.

Shape sensing with a multi-core fiber assumes that the distances betweencores in the fiber remain constant, when viewed in cross section,regardless of the shape of the fiber. This assumption is often validbecause glass is very hard and very elastic. Further, the cross sectionof the fiber (e.g., ˜125 microns) is small when compared with thedimensions of curves experienced by the fiber (e.g., bend radii greaterthan 5 mm). This maintenance of the cross-sectional position of thecores implies that all deformation of the fiber must be accommodated bythe elongation or the compression of the cores. As shown in FIG. 2 ,when a shape fiber is bent, a core on the outside 7 of the bend will beelongated while a core on the inside 8 of the bend will experiencecompression.

Since the average length of a fiber core segment is assumed to remainunchanged, an exercise in geometry shows that the change in the pointingdirection, (i.e., a vector that describes the position of the centralaxis of the fiber segment), can be calculated based on the change in thecore lengths and the distance between the cores. Other effects, such asthe strain-optic coefficient, must be taken into account. The result isthat the change in pointing direction for a given segment of fiber isdirectly proportional to the difference in length changes in the coreswithin that segment.

FIG. 3 shows that the bend in the fiber θ is proportional to the strainc in the off-center cores, where s is the segment length, r is radius,and k is a constant. In order to eliminate tension and temperature fromthe measurement, a differential measurement between the cores is used.

$\begin{matrix}{{\Delta\theta} = {{k\left( \frac{d_{s_{2}} - d_{s_{1}}}{s} \right)} = {k\left( {\varepsilon_{2} - \varepsilon_{1}} \right)}}} & {{Eq}.1}\end{matrix}$

The above equation describes the angular change for a given fibersegment and how it relates to a change in strain. Moving to the nextsegment in the fiber, the angular change of the previous segment must beadded to the next change in angle for the next segment to calculate thecurrent pointing direction of the fiber. In two dimensions, all of theprevious angles can be accumulated to find the bend angle at anyparticular location along the fiber. FIG. 4 shows the bend angle at anypoint or location along the fiber can be determined by a summation ofall angles leading up to that point, e.g., θ₅=θ₁+θ₂+θ₃+θ₄+θ₅. If thereare errors in measuring the angles, these errors accumulate along thefiber and result in a total error. This error becomes greater the longerthe fiber, growing as the square root of the number of segments.

To avoid this accumulated angle measurement error, the inventorsconceived of directly measuring the change in length of a segment ratherthan measuring strain. Mathematically, the summation of angles thenbecomes the summation of the length changes along the fiber as shown inequation (2) where L corresponds to fiber length.

$\begin{matrix}{\theta = {{\sum{\Delta\theta}} = {{k{\sum\frac{\left( {d_{s2} - d_{s1}} \right)}{s}}} = \frac{k\left( {{\Delta L_{2}} - {\Delta L_{1}}} \right)}{s}}}} & {{Eq}.2}\end{matrix}$

Thus, the angle at any position Z along the fiber then becomes linearlyproportional to the difference between the total changes in length ofthe cores up to that position as shown in equation (3).

θ(z) ∝ ΔL₂(z)−ΔL₁(z)   Eq. 3

Therefore, if the total length change along the fiber can be accuratelytracked continuously, rather than summing each individual local changein strain, the angular error can be prevented from growing. Later, itwill be shown how it is possible to track the change in length of a coreto an accuracy better than 10 nm, and to maintain this accuracy over theentire length of the fiber. This level of accuracy yields 0.3 degrees ofangular accuracy with a 70 micron separation between cores and,theoretically, about 0.5% of fiber length position accuracy.

Unfortunately, the cumulative relationship defined in (3) does not holdin three dimensions. But most three dimensional shapes can be accuratelyrepresented as a succession of two dimensional curves, and in thepresence of small angular changes (<10 degrees), three dimensionalangles also have this simple cumulative relationship. As a consequence,this approach is useful to assess error contributions in threedimensions.

The insight provided by this geometric exercise is that the total lengthchange as a function of distance along the multi-core fiber is usedrather than local strain. In other words, relatively larger errors inthe measured local strain values can be tolerated as long as theintegral of the measured strain corresponding to the total length changeup to that point, remains accurate. Nanostrain accuracies are achievedwithout requiring extremely large signal-to-noise ratios as thedistances over which the nanostrains are calculated are relatively large(e.g., many centimeters such as 10-1000 cm). As explained later indescription, the tracking of the change in length can also be used toassess rotation along the length of the fiber allowing higher thanexpected accuracies to be achieved in the measurement of fiber roll, orrotational angle around the fiber's axis, as well.

Phase Tracking in Optical Fiber

As a sensor, optical fiber can provide spatially continuous measurementsalong its entire length. Continuous measurements are important becauseoptical phase shifts are used to provide very high resolutiondisplacement measurements. Later it is explained how the intrinsicscatter in the fiber can be used to achieve this measurement, but it isconceptually easier to begin the explanation with Fiber Bragg Gratings(FBGs). A Fiber Bragg Grating is a periodic modulation of the index ofrefraction of the fiber. Each period is about one half of the wavelengthof the light in the fiber. The vacuum wavelength of the light is about1550 nm, and its wavelength in the fiber is about 1000 nm. The period ofthe grating is therefore about 500 nm. Typically a Bragg grating is usedas a sensor by measuring its reflected spectrum. The Bragg gratingcondition is calculated using the equation below.

λ_(B)=2nλ  Eq. 4

In this equation, λ_(B) represents wavelength, n is the index ofrefraction of fiber, and A corresponds to the period of the grating. Ifit is assumed that the index of refraction remains constant, then thereflected wavelength is solely dependent on the period of the grating.As the fiber is strained, the period of the grating is distorted,creating a shift in the reflected wavelength. Thus, for a shift inwavelength, it is possible to derive the amount of strain that wasapplied to the fiber. The period of a Bragg grating is highly uniform,and it is convenient to model this periodicity as a sinusoidalmodulation. When represented as a sinusoid, distortions in the period ofthe grating can be described as phase shifts. To illustrate thisconcept, consider the example in FIG. 5 which shows that as a fibercontaining Bragg gratings is strained, a phase difference measured froma reference state begins to accumulate.

The depiction of a strained Bragg grating shown in FIG. 5 illustratesthe local changes in index of refraction as alternating white andhatched segments. Assuming an ideal Bragg grating, all of the periodsare identical, and the phase of the modulation pattern increaseslinearly moving along the grating. In other words, the rate of change ofthe phase with distance is inversely proportional to the period of thegrating. If a small portion of the grating is stretched, then the rateof change of the phase decreases in the stretched portion.

In FIG. 5 , the top pattern depicts an undistorted grating with aperfectly linear phase as a function of position. The lower shiftedpattern depicts a grating distorted due to strain. The bottom graphshows the difference in phase between the two gratings at each location.The distortion in the grating results in a phase shift in the reflectedsignal of the grating with respect to the original undistorted phase. Aphase shift of 90 degrees is illustrated. After the strained segment,the rate of change returns to the unstrained state. However, the phasein this region is now offset from the original phase by an amount equalto the total phase change in the strained segment. This phase offset isdirectly proportional to the actual length change of the optical fiber.

This illustration shows only fifteen periods of the grating. Since aperiod is 500 nm, this amounts to 7.5 um in length. Stretching the fiberto induce a 90 degree phase shift displaced the remaining unstrainedgratings by a quarter of a period, or 125 nm. A typical OpticalFrequency Domain Reflectometry (OFDR) measurement may have a spatialresolution on the order of 50 microns. In other words, each OFDR datapoint, or index, is separated by 50 um. So a distortion of 125 nmresults in only a small fraction of an OFDR index shift in the actualposition of the grating. While the 125 nm change in position is notdetectable itself, the 90 degree phase shift is relatively easilymeasured with an OFDR system.

OFDR can therefore be used to measure distortions within Bragg gratings,and instead of only measuring the rate-of-change of the phase (i.e.,wavelength), the absolute phase can be measured, and from the phase,distance changes at each segment along the fiber core. This is importantfor accurate shape measurements in a situation where the phase in thegrating is observed to have changed, while the position of the gratingshows no readily discernable change. Conventional optical fibermeasurement technologies treat the phase shift and the position asseparate effects.

One way to visualize the relationship between the phase shift andposition is to imagine that the phase of the optical signal isrepresented by the second hand on a clock, and that the location alongthe fiber in index is represented by the hour hand on a clock. FIG. 6illustrates a clock with no minute hand. Such a clock makes it difficultto determine the time to a resolution of one minute. But this clock isstill useful for timing both short duration events with the second handand long duration events with the hour hand. Lacking a minute hand, itis not useful for measuring intermediate midscale duration events (e.g.,1 hour 12 minutes and 32 seconds) to one second precision. Thisdifficulty of linking the two scales has caused conventional opticalmeasurement systems to treat the phenomena separately.

This clock analogy helps to clarify why a continuous measurement isneeded along the entire length of the fiber. By monitoring the positionof the second hand continuously, the number of complete revolutions canbe measured, which allows the simultaneous monitoring of long durationsto a high precision. Linking the clock analogy to the previousdiscussion of Bragg gratings, each 360 degrees, or 2π, of phase changeequates to a 500 nm shift in location. By continuously tracking phasealong the optical fiber, both local strains and overall length changesof the optical fiber can be measured to a very high precision.

A challenge in tracking the phase continuously is that the resolution ofthe measurement must be sufficient such that the phase does not changefrom one segment to the next by more than 2π. FIG. 7 illustrates howthis lack of resolution can be problematic because there is no way todistinguish, for example, between a change of π/3 and a change ofπ/3+2π. So two different phase shifts will appear to have the same valueon the unit circle. In other words, an error of one index would beincurred in a count of full 2π revolutions. In this example, measurementof the overall change in length of the optical fiber would be deficientby 500 nm.

So it is it is important that a shape sensing system has sufficientresolution to guarantee the ability to track phase along the entirelength of a shape sensing fiber to ensure the accuracy of a shapesensing system.

Rayleigh Scatter-Based Measurements

As explained above, the typical use of an FBG for sensing involvesmeasuring shifts in the reflected spectrum of individual Bragg gratingsspaced at some interval down a fiber. Strain is derived for each sectionof fiber from the measurement for each Bragg grating. For shape sensingusing FBGs, each strain measurement indicates how much a given segmentis bent and in which direction. This information is summed for allmeasured segments to give the total fiber position and/or shape.However, using this method, an error in each segment accumulates alongthe fiber. The longer the fiber, the larger the error in themeasurement. This error using multiple Bragg gratings limits the speedof operation and the range of applications.

If there were a continuous grating along the fiber, then the phase couldbe tracked at every point along the fiber as described above. Trackingthe phase along the entire length of the core avoids accumulating error.Instead of accumulating error as the square root of the number of fibersegments, the total length error remains constant at a fraction of theoptical wavelength in the material. As mentioned earlier, a wavelengthof light can be about 1550 nm in a vacuum and about 1000 nm in thefiber, which is effectively 500 nm in reflection. A signal-to-noiseratio of 50 provides for an accuracy of 10 nm due to the round trip(reflective) nature of the measurement. The resulting strain accuracyover one meter of fiber will be 10 nanostrain.

Rayleigh scatter can be viewed as a Bragg grating with random phases andamplitudes or a Bragg grating consisting entirely of defects. ThisRayleigh scatter pattern, while random, is fixed within a fiber corewhen that core is manufactured. Strain applied to an optical fibercauses shifts or distortions in the Rayleigh scatter pattern. Theseinduced distortions of the Rayleigh scatter pattern can be used as ahigh resolution strain measurement for shape sensing by comparing areference scan of the fiber when the fiber is in a known shape with anew scan of the fiber when it has been bent or strained.

FIG. 8 shows example results of such a comparison. This figure shows thephase difference of the Rayleigh scatter signal between a reference scanand a measurement scan at the beginning of a section of fiber thatenters a region that is under tension. The data is plotted as a functionof fiber index, which represents distance along the fiber. Once theregion of tension is entered, the phase difference begins to accumulate.Since π and −π have the same value on the unit circle, the signalexperiences “wrapping” every multiple of 2π as the phase differencegrows along the length of the fiber. This can be seen around index 3350where the values to the left of this are approaching π, and thensuddenly the values are at −π. As shown, each wrap represents about 500nm of length change in the fiber. Since an index represents about 50microns of length, it takes about one hundred wraps of the phase toaccumulate a full index of delay change between measurement andreference.

The data in FIG. 9 is from the same data set as that for FIG. 8 , butfrom an area further down the fiber after about 35 wraps of the phase,or, roughly one third of an index. The noise on the phase differencedata has increased and is caused by the increasing shift between thereference and measurement scatter patterns. This decreases the coherencebetween the reference and measurement data used to determine the phasedifference. If the apparent location of an individual scattering fibersegment shifts by more than an index, then the coherence between thereference and the measurement is lost, and no strain measurement can beobtained from the comparison of scatter signals.

Therefore, the reference data should be matched to the measurement databy accounting for the shifting due to strain along the fiber. In thecase of one index being about 50 microns, over a one meter segment, thisamounts to only 50 parts per million, which is not a large strain. Infact, the weight of the fiber itself can induce strains on this order.Also, a change in temperature of only a few degrees Celsius can induce asimilar shift. Therefore, this shift in index should be accounted for inthe calculation of the distortion of the core.

A shift as a result of tension is a physical expansion of the individualsegments which results in an increased time of flight of the scatteredlight. The shift between reference and measurement is referred to asdelay. The delay can be accounted for by looking at a model of how ashift in the delay to any point in the sensing core affects the signalreflected from this point. If a field (light) is oscillating at afrequency, v, and it undergoes a delay of τ, then the optical phase as afunction of delay is given by,

ϕ=2πτv   Eq. 5

If the optical phase, ϕ, is plotted as a function of frequency, v, astraight line is obtained that intersects the origin. In practice,passing through a material such as glass distorts this curve from aperfect line, which should be kept in mind when comparing measuredvalues to the values predicted by this model. But for immediatepurposes, this model is sufficient. FIG. 10 shows this phase for twodifferent delays. In an example, non-limiting measurement system usingthe principle described above, a typical sweep of the laser might covera range of 192.5 to 194.5 THz. These frequencies represent a sweep from1542 nm (194.5 THz) to 1558 nm (192.5 THz), which has been a test sweeprange for a non-limiting, test shape sensing application. Over thisrange of interest, the phase for a given delay sweeps over a range ofΔϕ. For the two delays shown, τ₁ and τ₂, the difference in this sweeprange, Δϕ₂−Δϕ₁, is less than the change in phase at the centerfrequency, (193.5 THz), labeled dϕ. The factor between the change inphase at the center frequency and the change in phase sweep range willbe the ratio of the center frequency to the frequency sweep range. Inthis example case, the ratio is 96.7.

In the example test application, the sweep range, Δv, determines thespatial resolution, δτ, of the measurement. In other words, itdetermines the length of an index in the time domain. These are relatedby an inverse relationship:

δτ=1/(Δv)   Eq. 6

For the example frequency range described above, the length of an indexis 0.5 ps, or 50 microns in glass. At the center frequency, a phaseshift of 2π is induced by a change in delay of only 0.00516 ps, or 516nm in glass. A phase shift of 2π, then, represents only a fractionalindex shift in the time domain data. In order to shift the delay by oneindex in the time domain, the delay must change enough to induce a phasechange at the center frequency of 96.7×2π.

These examples illustrate that a linear phase change represents a shiftin the location of events in the time, or delay, domain. As seen above,a shift of one index will completely distort the measurements of phasechange along the length of the fiber. To properly compare the phases,then, these shifts should be accounted for as they happen, and thereference data should be aligned with the measurement data down theentire length of the core. To correct for this degradation of coherence,a temporal shift of the reference data is required. This may beaccomplished by multiplying the reference data for a given segment,r_(n), by a linear phase. Here n represents the index in the timedomain, or increasing distance along the fiber. The slope of this phasecorrection, γ, is found by performing a linear fit on the previous delayvalues. The phase offset in this correction term, φ, is selected suchthat the average value of this phase is zero.

{tilde over (r)} _(n) =r _(n) e ^(i(γn+φ))   Eq. 7

FIG. 11 shows the corrected phase difference over a section of fiberwhere a third of an index shift has occurred. The phase difference atthis location maintains the same signal-to-noise ratio as the closerpart of the fiber. By applying a temporal shift based on the delay at aparticular distance, coherence can be recovered reducing phase noise.

Example Shape Sensing Fiber

Tracking distortions in the Rayleigh scatter of optical fiber provideshigh resolution, continuous measurements of strain. The geometry of themulti-core shape sensing fiber is used to explain how this multi-corestructure enables measurements of both bend and bend direction along thelength of the fiber.

The optical fiber contains multiple cores in a configuration that allowsthe sensing of both an external twist and strain regardless of benddirection. One non-limiting, example embodiment of such a fiber is shownin FIG. 1 and described below. The fiber contains four cores. One coreis positioned along the center axis of the fiber. The three outer coresare placed concentric to this core at 120 degree intervals at aseparation of 70 um. The outer cores are rotated with respect to thecenter core creating a helix with a period of 66 turns per meter. Anillustration of this helically-wrapped multi-core shape sensing fiber isdepicted in FIG. 12 . A layout of a non-limiting test multi-core opticalfiber used in this discussion is pictured in FIG. 13 .

Another non-limiting example of a shape sensing fiber contains more thanthree outer cores to facilitate manufacture of the fiber or to acquireadditional data to improve system performance.

In a cross-section of a helixed fiber, the position of each outer coreappears to rotate around the center core progressing down the length ofthe fiber as illustrated in FIG. 14 .

Wobble Corection in Twisted Fiber

To translate strain signals from the outer cores in to bend and benddirection, the rotational position of an outer core must be determinedwith a high degree of accuracy. Assuming a constant spin rate of thehelix (see FIG. 12 ), the position of the outer cores may be determinedbased on the distance along the fiber. In practice, the manufacture ofhelixed fiber introduces some variation in the intended spin rate. Thevariation in spin rate along the length of the fiber causes an angulardeparture from the linear variation expected from the nominal spin rate,and this angular departure is referred to as a “wobble” and symbolizedas a wobble signal W(z).

One example test fiber manufactured with a helical multi-core geometryhas a very high degree of accuracy in terms of the average spin rate, 66turns per meter. However, over short distances (e.g., 30 cm) the spinrate varies significantly, and can cause the angular position to vary asmuch as 12 degrees from a purely linear phase change with distance. Thiserror in the spin rate is measured by placing the fiber in aconfiguration that will cause a continuous bend in a single plane, as isthe case for a coiled fiber on a flat surface. When the fiber is placedin such a coil, a helical core will alternate between tension andcompression as it travels through the outside portion of a bend and theinside portion of a bend. If phase distortion is plotted verse distance,a sinusoidal signal is formed with a period that matches the spin rateof the fiber. Variations in the manufacture of the multi-core fiber canbe detected as small shifts in the phase from the expected constant spinrate of the fiber.

An example of these variations in the spin rate is shown in FIG. 15 .The solid curve is the phase data (bend signal) taken from a planarcoil, and the dotted line is a generated perfect sinusoid at the samefrequency and phase as the helix. Note that at the beginning of the datasegment the curves are in phase with aligned zero crossings. By themiddle of the segment, the solid curve has advanced slightly ahead ofthe dotted curve, but by the end of the data segment, a significantoffset is observed. If the DC component of the phase signal is removed,and a phase shift calculated, the difference between these two signalsis significant and somewhat periodic.

FIG. 16 shows an example Wobble signal, W(z), with a periodic variationfrom a manufactured spin rate along the length of a shape sensing fiber.The phase variation is shown as a function of length in fiber index. Theexample data set represents about three meters of fiber. On the order ofa third of a meter, a periodicity in the nature of the spin rate of thefiber is detected. Over the length of the fiber, a consistent averagespin rate of the fiber is produced, but these small fluctuations shouldbe calibrated in order to correctly interpret the phase data produced bythe multi-core twisted fiber. This measurement in the change in spinrate or “wobble” is reproducible and is important to the calculation ofshape given practical manufacture of fiber.

Twist Sensing in Multi-Core Fibers

Torsion forces applied to the fiber also have the potential to induce arotational shift of the outer cores. To properly map the strain signalsof the cores to the correct bend directions, both wobble and appliedtwist must be measured along the entire length of the shape sensingfiber. The geometry of the helixed multi-core fiber enables directmeasurement of twist along the length of the fiber in addition tobend-induced strain as will be described below.

If a multi-core fiber is rotated as it is drawn, the central core isessentially unperturbed, while the outer cores follow a helical pathdown the fiber as shown in the center of FIG. 17 . If such a structureis then subjected to torsional stress, the length of the central coreremains constant. However, if the direction of the torsional stressmatches the draw of the helix, the period of the helix increases and theouter cores will be uniformly elongated as shown at the top of FIG. 17 .Conversely, if the torsional direction is counter to the draw of thehelix, the outer cores are “unwound” and experience a compression alongtheir length as shown at the bottom of FIG. 17 .

To derive the sensitivity of the multi-core configuration to twist, thechange in length that an outer core will experience due to torsion isestimated. A segment of fiber is modeled as a cylinder. The length L ofthe cylinder corresponds to the segment size, while the distance fromthe center core to an outer core represents the radius r of thecylinder. The surface of a cylinder can be represented as a rectangle ifone slices the cylinder longitudinally and then flattens the surface.The length of the surface equals the segment length L while the width ofthe surface corresponds to the circumference of the cylinder 2πr. Whenthe fiber is twisted, the end point of fiber moves around the cylinder,while the beginning point remains fixed. Projected on the flattenedsurface, the twisted core forms a diagonal line that is longer than thelength L of the rectangle. This change in length of the outer core isrelated to the twist in the fiber.

FIG. 18 shows an outer core that experiences twist can be modeled as aflattened cylinder as it translates along the surface. From the aboveflattened surface, the following can be shown:

$\begin{matrix}{{\partial d} \approx {\frac{2\pi r^{2}}{L}{\partial\phi}}} & {{Eq}.8}\end{matrix}$

In the above equation, ad is the change in length of the outer core dueto the change in rotation, ∂ϕ, of the fiber from its original helixedstate. The radial distance between a center core and an outer core isrepresented by r, and

$\frac{2\pi}{L}$

is the spin rate of the helical fiber in rotation per unit length.

The minimum detectable distance is assumed in this example to be a tenthof a radian of an optical wave. For the example test system, theoperational wavelength is 1550 nm, and the index of the glass is about1.47, resulting in a minimum detectable distance of approximately 10 nm.If the radius is 70 microns and the period of the helix is 15 mm, thenequation (8) indicates that the shape sensing fiber has a twistsensitivity of 0.3 deg. If the sensing fiber begins its shape byimmediately turning 90 degrees, so that the error due to twist weremaximized, then the resulting position error will be 0.5% of the fiberlength. In most applications, 90 degree bends do not occur at thebeginning of the fiber, and therefore, the error will be less than 0.5%.

Calculating Twist in a Four Core Fiber

The sensitivity of the twist measurement is based on the sensitivity ofa single core, but the sensing of twist along the length of the fiber isdependent on all four cores. If the difference in the change in thelength between the average of the outer cores and the center core isknown, then the twist (in terms of the absolute number of degrees)present in the fiber can be calculated.

The external twist along the fiber may be calculated using non-limiting,example procedures outlined in the flow chart shown in FIG. 19 . Thephase signals for all four cores A-D are determined, and the signals forouter cores B-D are averaged. The calculation of extrinsic twist isperformed by comparing the average of the outer core phase signals tothat of the center core. If the fiber experiences a torsional force, allouter cores experience a similar elongation or compression determined bythe orientation of the force to the spin direction of the helix. Thecenter core does not experience a change in length as a result of anapplied torsional force. However, the center core is susceptible totension and temperature changes and serves as a way of directlymeasuring common strain modes. Hence, if the center core phase signal issubtracted or removed from the average of the three outer cores, ameasure of phase change as a result of torsion is obtained. This phasechange can be scaled to a measure of extrinsic twist, or in other words,fiber rotation. Within the region of an applied twist over the length ofthe fiber less than a full rotation, this scale factor can beapproximated as linear. In the presence of high torsional forces, asecond order term should preferably be considered. Further, twistdistributes linearly between bonding points such that various regions oftwist can be observed along the length of the fiber.

FIG. 20 shows an example data set for a generic shape that illustratesthe FIG. 19 algorithm in more detail. The graph shows phase distortionas a result of local change in length of the center core (black) and anouter core (gray) of a shape sensing fiber for a general bend. The twophase curves shown in FIG. 20 represent the local changes in lengthexperience by two of the cores in the multi-core shape sensing fiber.The curves for two of the outer cores are not shown in an effort to keepthe graphs clear, but the values from these other two cores are used indetermining the final shape of the fiber.

The center core phase signal does not experience periodic oscillations.The oscillations are a result of an outer core transitioning betweencompressive and tensile modes as the helix propagates through a givenbend. The central core accumulates phase along the length of the shapesensing fiber even though it is not susceptible to bend or twist inducedstrain. The center core phase signal describes common mode strainexperienced by all cores of the fiber. The outer cores are averaged(gray) and plotted against the center core (black) in FIG. 21 .

As the outer cores are 120 degrees out of phase, the bend inducedvariation in the phase signals averages to zero. FIG. 21 , a slightdeviation between the two phase curves is observed. Subtracting thecenter core phase, a direct measure of common mode strain, leaves thephase accumulated as a result of torsional forces. With proper scaling,this signal can be scaled to a measure of fiber roll designated as the“twist” signal T(z) produced from FIG. 21 which is shown in FIG. 22 .From the twist signal, T(z), the shift in rotational position of theouter cores as a result of torsion along the length of the shape sensingfiber tether can be determined. This allows a bend signal to be mappedto the correct bend direction.

The desirability of compensating for twist in the shape calculation isillustrated by the data set shown in FIG. 23 . The tip of the shapesensing fiber was translated in a single plane through a five point gridforming a 250 mm square with a point at its center with shape processingconsidering twist (filled). The correction for external twist was notused in the processing of the data set plotted as unfilled dots. In theplot, it is impossible to distinguish the original shape traced with thetip of the fiber if the twist calculation is not used. Even for smallfiber tip translations, significant twist is accumulated along thelength of the fiber. Thus, if this twist is not accommodated for in theshape sensing, then significant levels of accuracy cannot be achieved.

Calculation of Bend Induced Strain

Along with information describing the amount of twist applied to theshape sensing fiber, a multi-core fiber also enables extraction of bendinformation in an ortho-normal coordinate system. The phase signals forfour optical cores of the shape sensing fiber can be interpreted toprovide two orthogonal differential strain measurements as describedbelow. These strain values can then be used to track a pointing vectoralong the length of the fiber, ultimately providing a measure of fiberposition and/or shape.

With the common mode strain removed, the three, corrected, outer corephase signals are used to extract a measure of bend along the shapesensing fiber. Due to symmetry, two of the outer cores can be used toreconstruct the strain signals along the length of the fiber. First, thederivative of the phase signal for two of the outer cores is taken. Thisderivative is preferably calculated so that the error on the integral ofthe derivative is not allowed to grow, which translates to a loss inaccuracy of the system. For double-precision operations, this is not aconcern. But if the operations are done with a limited numericprecision, then rounding must be applied such that the value of theintegral does not accumulate error (convergent rounding).

Assume for this explanation that strain can be projected in a linearfashion. Thus, the phase response of a given core is a combination oftwo orthogonal strains projected against their radial separation.

$\begin{matrix}{\frac{d{\phi}_{n}}{dz} = {{{b_{y}(z)}{\sin\left( {{kz} + \Delta_{n}} \right)}} + {{b_{x}(z)}{\cos\left( {{kz} + \Delta_{n}} \right)}}}} & {{Eq}.9}\end{matrix}$

In the above equation, b_(x) and b_(y) are the orthogonal strain signalsused to calculate bend. The phase, ϕ_(n), represents the phase responseof a core, z is the axial distance along the fiber, k is the spin rateof the helix, and the delta Δ represents the radial position of the core(120 degree separation).

The phase response from two of the outer cores is:

$\begin{matrix}{\frac{d\phi_{1}}{dz} = {{{b_{y}(z)}{\sin\left( {{kz} + \Delta_{1}} \right)}} + {{b_{x}(z)}{\cos\left( {{kz} + \Delta_{1}} \right)}}}} & {{Eq}.10}\end{matrix}$ $\begin{matrix}{\frac{d\phi_{2}}{dz} = {{{b_{y}(z)}{\sin\left( {{kz} + \Delta_{2}} \right)}} + {{b_{x}(z)}{\cos\left( {{kz} + \Delta_{2}} \right)}}}} & {{Eq}.11}\end{matrix}$

Solving for b_(x) and b_(y):

$\begin{matrix}{{b_{y}(z)} = {\frac{1}{\sin\left( {\Delta_{1} - \Delta_{2}} \right)}\left\lbrack {{\frac{d\phi_{1}}{dz}{\cos\left( {{kz} + \Delta_{2}} \right)}} + {\frac{d\phi_{2}}{dz}{\cos\left( {{kz} + \Delta_{1}} \right)}}} \right\rbrack}} & {{Eq}.12}\end{matrix}$ $\begin{matrix}{{b_{x}(z)} = {\frac{1}{\sin\left( {\Delta_{2} - \Delta_{1}} \right)}\left\lbrack {{\frac{d\phi_{1}}{dz}{\sin\left( {{kz} + \Delta_{2}} \right)}} + {\frac{d\phi_{2}}{dz}{\sin\left( {{kz} + \Delta_{1}} \right)}}} \right\rbrack}} & {{Eq}.13}\end{matrix}$

In the above equations 12 and 13, k, the spin rate, is assumed constantalong the length of the fiber. The above derivation remains valid ifcorrection terms are added to the spin rate. Specifically, the measuredwobble W(z) and twist signals T(z) are included to compensate for therotational variation of the outer cores along the length of the fiber.The above expressions (12) and (13) then become the following:

$\begin{matrix}{{b_{y}(z)} = {\frac{1}{\sin\left( {\Delta_{1} - \Delta_{2}} \right)}\left\lbrack {{\frac{d\phi_{1}}{dz}{\cos\left( {{kz} + {T(z)} + {W(z)} + \Delta_{2}} \right)}} + {\frac{d\phi_{2}}{dz}{\cos\left( {{kz} + {T(z)} + {W(z)} + \Delta_{1}} \right)}}} \right\rbrack}} & {{Eq}.14}\end{matrix}$ $\begin{matrix}{{b_{x}(z)} = {\frac{1}{\sin\left( {\Delta_{1} - \Delta_{2}} \right)}\left\lbrack {{\frac{d\phi_{1}}{dz}{\sin\left( {{kz} + {T(z)} + {W(z)} + \Delta_{2}} \right)}} + {\frac{d\phi_{2}}{dz}{\sin\left( {{kz} + {T(z)} + {W(z)} + \Delta_{1}} \right)}}} \right\rbrack}} & {{Eq}.15}\end{matrix}$

Calculation of Shape from Orthogonal Differential Strain Signals

Equations (14) and (15) produce two differential, orthogonal strainsignals. FIG. 24 depicts the orthogonal strain curves for a fiber placedin several bends that all occur in the same plane. These twodifferential, orthogonal strain signals are processed to perform thefinal integration along the length of the shape sensing fiber to producethree Cartesian signals representing the position and/or shape of thefiber.

FIG. 25 shows a flowchart diagram describing non-limiting, example stepsfor calculating shape from strain. Orthonormal strain signals A and Bare determined according to the equations 14 and 15.

The acquired data at the data acquisition network is preferably storedin discrete arrays in computer memory. To do this, a change inrepresentation from the continuous representation in equation 15 to adiscrete representation based on index is needed at this point. Further,the bend at each point in the array can be converted to an angularrotation since the length of the segment (Δz) is fixed and finite usingequation (1). The parameter, a, is determined by the distance of thecores from the center of the fiber and the strain-optic coefficientwhich is a proportionality constant relating strain to change in opticalpath length.

θ_(y,n)=ab_(y,n)Δz   Eq. 16

θ_(x,n)=ab_(x,n)Δz   Eq. 17

These measures of rotation θ due to local bend in the fiber can be usedto form a rotation matrix in three dimensions. If one imagines beginningwith the fiber aligned with the z axis, the two bend components rotatethe vector representing the first segment of the fiber by these twosmall rotations. Mathematically, this is done using a matrixmultiplication. For small rotations, the simplified rotation matrixshown in equation (18) below can be used.

$\begin{matrix}{\overset{\_}{\overset{\_}{R_{n}}} = \begin{bmatrix}1 & 0 & \theta_{x,n} \\0 & 1 & \theta_{y,n} \\{- \theta_{x,n}} & \theta_{y,n} & 1\end{bmatrix}} & {{Eq}.18}\end{matrix}$

The above rotation matrix is valid if θ_(x)<<1 and θ_(y)<<1. If theresolution of the system is on the order of micrometers, this is acondition that is not difficult to maintain. After rotation, the fibersegment will have a new end point and a new direction. All further bendsare measured from this new pointing direction. Therefore, the pointingdirection (or vector) at any position on the fiber depends upon all ofthe pointing directions between that location in the fiber and thestarting location. The pointing vector at any point of the fiber can besolved in an iterative process tracking the rotational coordinate systemalong the length of the fiber as seen in the following expression:

C _(n+1) = C _(n) R _(n)   Eq. 19

In other words, each segment along the fiber introduces a small rotationproportional to the size and direction of the bend along that segment.This iterative calculation can be written in mathematical notationbelow:

C _(p) = C ₀ Π_(n=0) ^(p) R _(n)   Eq. 20

Here again, for small rotations and nearly planar rotations, the anglesare effectively summed, and by maintaining an accurate measure of theintegral of the strain (the length change) throughout the length of theshape sensing fiber, better accuracy is achieved than is possible usingthe strain alone. The matrix calculated above contains information aboutthe local orientation of the cores, which allows for the properrotations to be applied. If the primary interest is in determining theposition along the fiber, then only the local vector that describes thepointing direction of the fiber at that location is needed. Thispointing vector can be found by a simple dot product operation.

$\begin{matrix}{P = {\overset{\_}{\overset{\_}{C}} \cdot \begin{bmatrix}0 \\0 \\1\end{bmatrix}}} & {{Eq}.21}\end{matrix}$

If each of these pointing vectors is placed head-to-tail, as illustratedin the FIG. 26 , an accurate measurement of the shape results. Thus, theposition and/or direction at any point along the length of the fiber canbe found by the summation of all previous pointing vectors, scaled tothe resolution of the system:

$\begin{matrix}{\begin{bmatrix}x \\y \\z\end{bmatrix}_{p} = {\Delta d{\sum_{p = 0}^{q}\left\lbrack {\left\{ {\overset{\_}{\overset{\_}{C_{0}}}{\prod_{n = 0}^{p}\overset{\_}{\overset{\_}{R_{n}}}}} \right\} \cdot \begin{bmatrix}0 \\0 \\1\end{bmatrix}} \right\rbrack}}} & {{Eq}.22}\end{matrix}$

One non-limiting example of a shape sensing system is described inconjunction with FIG. 27 . Other implementations and/or components maybe used. Moreover, not every component shown is necessarily essential.The System Controller and data processor (A) initiates two consecutivesweeps of a tunable laser (B) over a defined wavelength range and tuningrate. Light emitted from the tunable laser is routed to two opticalnetworks via an optical coupler (C). The first of these two opticalnetworks is a Laser Monitor Network (E) while the second is designatedas an Interrogator Network (D). Within the Laser Monitor Network (E),light is split via an optical coupler (F) and sent to a gas (e.g.,Hydrogen Cyanide) cell reference (G) used for C-Band wavelengthcalibration. The gas cell spectrum is acquired by a photodiode detector(L) linked to a Data Acquisition Network (U).

The remaining portion of light split at optical coupler (F) is routed toan interferometer constructed from an optical coupler (H) attached totwo Faraday Rotator Mirrors (I,J). The first Faraday Rotator Mirror(FRMS) (I) serves as the reference arm of the interferometer while thesecond Faraday Rotator Mirror (J) is distanced by a delay spool (K) ofoptical fiber. This interferometer produces a monitor signal that isused to correct for laser tuning nonlinearity and is acquired by theData Acquisition Network (U) via a photodiode detector (M).

Light routed to the Interrogator Network (D) by optical coupler (C)enters a polarization controller (N) that rotates the laser light to anorthogonal state between the two successive laser scans. This light isthen split via a series of optical couplers (O) evenly between fouracquisition interferometers (P, Q, R, S). Within the acquisitioninterferometer for the central core, light is split between a referencepath and a measurement path by an optical coupler (AA). The “probe”laser light from coupler AA passes through an optical circulator (T) andenters a central core of a shape sensing fiber (W) through a centralcore lead of a multi-core fanout (V) for the shape sensing fiber (W).The shape sensing fiber (W) contains a central optical core concentricto three helically wound outer optical cores. The cross section of thefiber (X) depicts that the outer cores (Z) are evenly spaced,concentric, and separated by a given radial distance from the centralcore (Y). The resulting Rayleigh backscatter of the central optical core(Y) as a consequence of a laser scan passes through the opticalcirculator (T) and interferes with the reference path light of theacquisition interferometer when recombined at optical coupler (BB).

The interference pattern passes through an optical polarization beamsplitter (DD) separating the interference signal into the two principlepolarization states (S₁, P₁). Each of the two polarization states isacquired by the Data Acquisition Network (U) using two photodiodedetectors (EE, FF). A polarization rotator (CC) can be adjusted tobalance the signals at the photodiode detectors. The outer optical coresof the shape sensing fiber are measured in a similar manner usingcorresponding acquisition interferometers (Q, R, S). The SystemController and Data Processor (A) interprets the signals of the fourindividual optical cores and produces a measurement of both position andorientation along the length of the shape sensing fiber (W). Data isthen exported from the System Controller (A) for display and/or use(GG).

Birefringence Corrections

When an optical fiber is bent, the circular symmetry of the core isbroken, and a preferential “vertical” and “horizontal” is created by thedistinction between directions in the plane of the bend andperpendicular to the plane of the bend. Light traveling down the fiberthen experiences different indices of refraction depending upon itspolarization state. This change in the index as a function ofpolarization state is referred to as birefringence. This presents asignificant problem for shape measurement because the measured phasechange depends on the incident polarization state, and this incidentstate cannot be controlled in standard fiber.

This problem can be solved by measuring the optical core response at twoorthogonal polarization states. If the response of these two states isaveraged properly, the variation in the measured response as a functionof polarization can be eliminated or at least substantially reduced. Theflowchart diagram in FIG. 28 outlines a non-limiting, example processfor correcting for birefringence such as intrinsic birefringence,bend-induced birefringence, etc. both in measured and in referencevalues. The non-limiting example below relates to bend-inducedbirefringence but is more generally applicable to any birefringence.

The first step in the process is to measure the response of the core attwo orthogonal polarization states called “s” and “p”. An s response andap response are measured at each polarization state resulting in fourarrays. For simplicity, the responses to the first polarization stateare called a and b, and the responses to the second polarization stateare called c and d, where a and c are the responses at the s detectorand b and d are the responses at the p detector.

The second step is to calculate the following two array products:

x=ad*   Eq. 23

y=bc*   Eq. 24

A low-pass filtered version of each of these signals is calculated whichis written as, <x> and <y>. The expected value notation is used here toindicate a low-pass filtering operation. The phases of the relativelyslowly varying functions are used to align the higher frequency scattersignals in phase so that they can be added:

p=a+d

  Eq. 25

q=b+c

  Eq. 26

This process is then repeated to produce a final scalar value:

u=p+qe^(i∠<pq*>)  Eq. 27

Now, a slowly varying vector can be created that represents the vectornature of the variation down the fiber without wideband Rayleigh scattercomponents, since these are all subsumed into u:

v =[

ae ^(i∠u*)

,

be ^(i∠u*)

,

ce ^(i∠u*)

,

de ^(i∠u*)

]  Eq. 28

The correction due to birefringence effects is then calculated using:

ϕ_(n)=∠( v _(n) ·v ₀*)   Eq. 29

where ϕ_(n) is the correction due to birefringence effects and n is theindex into the array. Here the vector is shown compared to the firstelement (index 0) in the array, but it can just as easily be comparedwith any arbitrarily selected element in the vector array.

The birefringence correction compensates for birefringence as result ofcore asymmetry during manufacture and for bend radii in excess of 100mm. As the shape sensing fiber is placed into tight bends with radiiless than 100 mm, a second order birefringence effect becomessignificant.

Assuming that significant levels of strain only manifest in thedirection parallel to the central core of the multi-core shape sensingfiber, consider the diagram in FIG. 29 . As the fiber is bent, tensilestrain is measured in the region between 0<X≤r while compressive strainis measured in the region −r≤X<0. The expansion of the outer bend regionexerts a lateral force increasing the internal pressure of the fiber. Asthe internal pressure of the fiber increases, a second order strain termbecomes significant, ε_(x). As shown in the second graph, this pressurestrain term is a maximum along the central axis of the fiber and fallsoff towards the outer edges of the fiber as the square of distance. Intight bends, this pressure strain term can modify the index ofrefraction of the fiber resulting in measureable birefringence. Further,the outer peripheral helical cores experience a sinusoidal response tothis pressure induced strain while the center core responds to themaximum.

FIG. 30 shows two phase plots produced from a 40 mm diameter fiber loop.Oscillations in these signals are a result of the multi-core assemblybeing off center of the fiber. In tighter bends, strain signals are highenough to elicit a response from this subtle deviation fromconcentricity. The plot shows that the average of the helical outercores accumulates significantly less phase in the region of the bendwhen compared to the center core. This phase deficiency serves asevidence for bend induced birefringence. Recall that the extrinsic twistcalculation is performed by finding the absolute phase differencebetween the center core and the average of the three outer cores. Thegraph in FIG. 30 shows that a false twist signal will be measured in theregion of the bend.

The measured phase response of an outer core indicates its positionrelative to the pressure-induced strain profile, ε_(x). Therefore, thesquare of an outer core strain response provides a measure of bothlocation and magnitude relative to the pressure field. This response maybe scaled and used as a correction to the outer cores to match the levelof Ex perceived by the central core, thereby correcting for the falsetwist.

$\begin{matrix}{\phi_{ncorr} = {\phi_{n} - {k{\int\left\lbrack \left( {\frac{d\phi_{n}}{dz} - \frac{\sum_{i = 0}^{N}\frac{d\phi_{n}}{dz}}{N}} \right)^{2} \right\rbrack}}}} & {{Eq}.30}\end{matrix}$

φ_(n) is the phase response on an outer core, N is the number of outercores, and k serves as a scale factor. FIG. 31 shows the strain responseof an outer core for a 40 mm diameter fiber loop, with common modestrain subtracted. From this strain response signal a correction forbend induced birefringence can be approximated as is seen in the graphshown in FIG. 32 .

Applying this correction has a significant impact on the measured twistin the region of the bend as shown in FIG. 33 . Comparing the twistsignal with and without 2^(nd) order correction reveals that a 25 degreeerror is accumulated in the bend region without the 2^(nd) orderbirefringence correction in this example.

Applying Birefringence Corrections and Impact on Accuracy

The following describes the effects of polarization on the accuracy of ashape sensing system. To achieve a varying input polarization betweenmeasurements, a loop polarization controller is added between the shapesensing fiber and the shape sensing system as illustrated in FIG. 34 .

To illustrate the impact of the above-described corrections on theaccuracy of the system, consider the in-plane signal for a relativelysimple shape as shown in FIG. 35 , where 1.4 meters of shape sensingfiber is routed through a single 180 degree turn with a bend radius of50 mm. FIG. 36 shows out-of-plane measurements for three successivemeasurements. Between each measurement, the polarization is varied usingthe polarization controller in FIG. 34 .

If birefringence is not considered, a significant loss in accuracy andprecision is observed. A large response is observed in the out-of-planesignal as the polarization state is varied. The fiber picks up anangular error only in the region of the bend as a result of the systemmeasuring an erroneous twist signal. Thus, when exiting this bend, thereis a significant error in the pointing direction of the fiber.Predicting the polarization response of the fiber is a difficultproblem, and not every core responds to the same extent for a givenbend. FIG. 37 illustrates this point showing the birefringencecorrections for cores. However, the same two measurements for the centercore have a significant variation in their phase responses as seen inFIG. 38 . Two successive measurements respond differently to inputpolarization providing evidence for birefringence in the shape sensingfiber.

Activating a correction for birefringence improved the precision of thesystem as seen in FIG. 39 . The variation between shape measurements asthe input polarization state varies is minimized which greatly increasesthe precision of the system. However, a significant error in theaccuracy of the system is still observed. If the second order correctionbased on bend induced birefringence is also performed, there is furtherimprovement of the system as shown in FIG. 40 . Both the precision andaccuracy of the out of plane signal are dramatically improved.

Although various embodiments have been shown and described in detail,the claims are not limited to any particular embodiment or example. Noneof the above description should be read as implying that any particularelement, step, range, or function is essential such that it must beincluded in the claims scope. The scope of patented subject matter isdefined only by the claims. The extent of legal protection is defined bythe words recited in the allowed claims and their equivalents. Allstructural and functional equivalents to the elements of theabove-described preferred embodiment that are known to those of ordinaryskill in the art are expressly incorporated herein by reference and areintended to be encompassed by the present claims. Moreover, it is notnecessary for a device or method to address each and every problemsought to be solved by the present invention, for it to be encompassedby the present claims. No claim is intended to invoke paragraph 6 of 35USC § 112 unless the words “means for” or “step for” are used.Furthermore, no embodiment, feature, component, or step in thisspecification is intended to be dedicated to the public regardless ofwhether the embodiment, feature, component, or step is recited in theclaims.

What is claimed is:
 1. A shape-sensing system for a multi-core opticalfiber, the system comprising: measurement means for measuring reflectedlight signals of a plurality of cores of the optical fiber; andprocessing means for processing the measured reflected light signals to:continuously track, for each core of the plurality of cores, a phasedifference accumulated along the optical fiber, the phase differencebeing between phases of the measured reflected light signals andreference reflected light signals, and based on the continuously trackedphase differences for the plurality of cores, determine a shape of theoptical fiber.
 2. The shape-sensing system of claim 1, wherein: themeasurement means comprise a plurality of acquisition interferometersand associated data acquisition networks; and the processing meanscomprise a system controller and data processor configured for opticalfrequency domain reflectometry (OFDR).
 3. The shape-sensing system ofclaim 1, further comprising: tunable laser means for emitting light overa range of wavelengths; and optical routing means for routing theemitted light to the plurality of cores and causing generation of thereflected light signals in the plurality of cores.
 4. The shape-sensingsystem of claim 1, wherein the measurement means comprise: means tointerferometrically combine the reflected light signals with referencepath signals to generate interference signals for the plurality ofcores; and means to separate the interference signals into a firstpolarization state and a second polarization state orthogonal to thefirst polarization state.
 5. The shape-sensing system of claim 4,wherein the measurement means further comprise: first detecting meansfor detecting the first polarization state; and second detecting meansfor detecting the second polarization state.
 6. The shape-sensing systemof claim 1, wherein the processing means comprise: alignment means foraligning the measured reflected light signals with the referencereflected light signals to account for a shift in apparent location ofsegments of the optical fiber between a measurement state and areference state.
 7. The shape-sensing system of claim 1, wherein theprocessing means comprise: a system controller and data processorconfigured to determine the shape of the optical fiber by using thecontinuously tracked phase differences to track a pointing vector alongthe optical fiber.
 8. The shape-sensing system of claim 1, wherein theprocessing means comprise determination means for determining the shapeof the optical fiber by: computing first and second orthogonaldifferential strain signals from derivatives of the continuously trackedphase differences for first and second cores of the plurality of cores;using a discrete representation of the first and second orthogonaldifferential strain signals to compute rotation matrices for segmentsalong the optical fiber; and computing local pointing vectors from therotation matrices.
 9. The shape-sensing system of claim 8, wherein thedetermination means comprise: summing means for determining a positionand direction at a point along the optical fiber by summing all localpointing vectors up to that point.
 10. A shape-sensing system for amulticore optical fiber comprising a center core and two or more helixedouter cores, the system comprising: measurement means for measuringlight reflected in the center core and the two or more helixed outercores of the optical fiber; tracking means for tracking, based on themeasured light, phases associated with strain in the center core and thetwo or more helixed outer cores along a length of the optical fiber;determination means for determining rotational positions of the two ormore helixed outer cores along the length of the optical fiber; andcomputing means for computing a shape of the optical fiber based atleast in part on the tracked phases and the determined rotationalpositions.
 11. The shape-sensing system of claim 10, wherein thedetermination means comprise: means for determining a twist signalindicative of extrinsic twist applied to the optical fiber.
 12. Theshape-sensing system of claim 11, wherein means for determining thetwist signal comprises: subtracting means for subtracting the phaseassociated with strain in the center core from an average of the phasesassociated with strain in the two or more helixed outer cores.
 13. Theshape-sensing system of claim 10, wherein the determination meanscomprise: means for determining a wobble signal indicative of avariation in spin rate along the length of the optical fiber.
 14. Theshape-sensing system of claim 10, wherein the tracking means comprise:alignment means for aligning a reference scan of the multicore opticalfiber in a known shape against a measurement scan of the multicoreoptical fiber to maintain coherence, and comparison means for comparingthe reference scan with the measurement scan.
 15. The shape-sensingsystem of claim 10, wherein the computing means comprise means to:determine two orthogonal differential strain signals based at least inpart on the tracked phases and at least one signal selected from thegroup consisting of: a twist signal and a wobble signal indicative ofthe rotational positions; create a rotation matrix comprising bendangles determined based on the two orthogonal differential strainsignals; use the rotation matrix to iteratively compute pointing vectorsalong the length of the optical fiber; and compute the shape of theoptical fiber based on sums of the pointing vectors.
 16. Theshape-sensing system of claim 10, wherein: the measurement meanscomprise: interferometric means to interferometrically combine thereflected light with reference path light into interference signals forthe center core and the two or more helixed outer cores; separationmeans for separating the interference signals into two orthogonalpolarization states; and detecting means for detecting the twoorthogonal polarization states.
 17. A system for correcting forbirefringence while shape sensing with an optical fiber comprising aplurality of cores, the system comprising: rotation means to rotate apolarization of polarized light between a first polarization and asecond polarization; measurement means to, for each core of theplurality of cores, couple the polarized light into the core, create aninterference signal between light reflected from the core and referencepath light, and measure the interference signal in two principlepolarization states; computing means to compute, for each core of theplurality of cores, a response signal corrected for birefringence basedon measurements of the interference signal for the first polarizationmeasured at the two principle polarization states and the interferencesignal for the second polarization measured at the two polarizationstates; and determination means to determine a shape of the opticalfiber from the response signals corrected for birefringence for theplurality of cores.
 18. The system of claim 17, further comprising:scanning means to scan the polarized light across a range ofwavelengths, wherein the polarization is rotated between successivescans.
 19. The system of claim 17, wherein: the plurality of corescomprises a central core and a plurality of outer cores; and thedetermination means comprise: computing means to compute, for each coreof the plurality of outer cores, a second-order birefringence correctionbased on the response signals of cores of the plurality of outer cores.20. The system of claim 19, wherein the determination means comprise:means to use at least the second-order birefringence correction indetermining an extrinsic twist of the optical fiber.